Abstract
In this short note we give an overview of recent work on string propagation on stacks and applications to gauged linear sigma models. We begin by outlining noneffective orbifolds (orbifolds in which a subgroup acts trivially) and related phenomena in two-dimensional gauge theories, which realize string propagation on gerbes. We then discuss the 'decomposition conjecture,' equating conformal field theories of strings on gerbes and strings on disjoint unions of spaces. Finally, we apply these ideas to gauged linear sigma models for complete intersections of quadrics, and use the decomposition conjecture to show that the Landau-Ginzburg points of those models have a geometric interpretation in terms of a (sometimes noncommutative resolution of) a branched double cover, realized via nonperturbative effects rather than as the vanishing locus of a superpotential. These examples violate old unproven lore on GLSM's (namely, that geometric phases must be related by birational transformations), and we conclude by observing that in these examples (and conjecturing more generally in GLSM's), the phases are instead related by Kuznetsov's 'homological projective duality.'
Highlights
The GLSM realizes: Kahler branched double cover of P3 (Clemens’ octic double solid) where RHS realized at LG point via local Z2 gerbe structure + Berry phase
K’s noncomm’ res’n is defined by (P3,B) where B is the sheaf of even parts of Clifford algebras associated with the universal quadric over P3 defined by the GLSM superpotential
Kahler branched double cover of P3 where RHS realized at LG point via local Z2 gerbe structure + Berry phase
Summary
-- is a powerful tool, but we really can’t follow it completely explicitly in general. -- can’t really prove in any sense that two theories will flow under renormalization group to same point. We do lots of calculations, perform lots of consistency tests, and if all works out, we believe it. The problems here are analogous to the derivedcategories-in-physics program. There, to a given object in a derived category, one picks a representative with a physical description (as branes/antibranes/tachyons). It is conjectured that different representatives give rise to the same low-energy physics, via boundary renormalization group flow. Potential problems / reasons to believe that presentation-independence fails:
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